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A pseudo-distance on a set E is a mapping d of E x E into R which is such
that d(x, *) = 0 for all x e E, and satisfies axioms (I), (III), and (IV) of (3.1),
but not necessarily axiom (II). It is clear that a pseudo-distance satisfies the

d(xl9 JCM) ^ d(xl9 x2) 4- d(x2, x3) -}-••• + d(xn-l9 xn),
\d(x, z) — d(y, z)\ ^ d(x, y)

for all Xi and x, y, z in E.

(12.4,1)   If /is flfly mapping of a set E into R, the mapping

is a pseudo-distance.

Let <p be a mapping of the interval [0, -h oo [ into itself which satisfies the
following three conditions :

(1) <p(0) = 0; (2) q> is increasing; (3) (p(u + v)^ <p(u) + <p(v) for all u *z 0
and v ^ 0.

Then if rfis a pseudo-distance on a set E, the same is true of the composite
mapping (p ° d: (x, y)\-+<p(d(x, y)). We have only to check the triangle
inequality, and by conditions (2) and (3) we have

<p(d(x9 z)) £ <p(d(x, y) + d(y, z)) ^ <p(d(x9 y)) + ?(d[x, z)).

If (^/n) is a sequence of pseudo-distances on a set E, such that the series
x> y) converges for all (x, y) e E x E, then its sum d(x, y) is a pseudo-

distance on E (3.15.4).

Now consider a family (rfa)aei of pseudo-distances on a set E. For each
0eE, each finite family («/)igy£m of elements of I and each finite family
of real numbers >0, put

a,x)<rJ   for   l^ygw},
i                              flf,jf)gr>/   for   l^ygm}.

Let O denote the set of subsets U of E such that, for each x e U, there exists
a finite family (<Xj)igj*m of elements of I and a finite family (rj)igjgm of
strictly positive real numbers such that B(x; (a,-), (r^)) c U. It is immediately
verified that D is a topology on E. This topology is said to be defined by theoint open sets U, V, such that A c U and B cr V. (Consider